A heavy Wigner matrix 2123616303044&_mathId=si1.gif&_user=111111111&_pii=S0022123616303044&_rdoc=1&_issn=00221236&md5=7789e3b5f1c7aab33951f9ca642d1ce9" title="Click to view the MathML source">XN is defined similarly to a classical Wigner one. It is Hermitian, with independent sub-diagonal entries. The diagonal entries and the non-diagonal entries are identically distributed. Nevertheless, the moments of the entries of 2123616303044&_mathId=si2.gif&_user=111111111&_pii=S0022123616303044&_rdoc=1&_issn=00221236&md5=7131c4f93952d3061342a628e8a4ecfc">2123616303044-si2.gif"> tend to infinity with N , as for matrices with truncated heavy tailed entries or adjacency matrices of sparse Erdös–Rényi graphs. Consider a family 2123616303044&_mathId=si137.gif&_user=111111111&_pii=S0022123616303044&_rdoc=1&_issn=00221236&md5=dabafce5b4a8d546565fea5f999cdf88" title="Click to view the MathML source">XN of independent heavy Wigner matrices and an independent family 2123616303044&_mathId=si138.gif&_user=111111111&_pii=S0022123616303044&_rdoc=1&_issn=00221236&md5=bdb7ce74926a6d247eb9a741b02850dc" title="Click to view the MathML source">YN of arbitrary random matrices with a bound condition and converging in ⁎-distribution in the sense of free probability. We characterize the possible limiting joint ⁎-distributions of 2123616303044&_mathId=si139.gif&_user=111111111&_pii=S0022123616303044&_rdoc=1&_issn=00221236&md5=c585460f596fa94ab6e6f9a8ff92f82e" title="Click to view the MathML source">(XN,YN), giving explicit formulas for joint ⁎-moments. We find that they depend on more than the ⁎-distribution of 2123616303044&_mathId=si138.gif&_user=111111111&_pii=S0022123616303044&_rdoc=1&_issn=00221236&md5=bdb7ce74926a6d247eb9a741b02850dc" title="Click to view the MathML source">YN and that in general 2123616303044&_mathId=si137.gif&_user=111111111&_pii=S0022123616303044&_rdoc=1&_issn=00221236&md5=dabafce5b4a8d546565fea5f999cdf88" title="Click to view the MathML source">XN and 2123616303044&_mathId=si138.gif&_user=111111111&_pii=S0022123616303044&_rdoc=1&_issn=00221236&md5=bdb7ce74926a6d247eb9a741b02850dc" title="Click to view the MathML source">YN are not asymptotically ⁎-free. We use the traffic distributions and the associated notion of independence 210">[21] to encode the information on 2123616303044&_mathId=si138.gif&_user=111111111&_pii=S0022123616303044&_rdoc=1&_issn=00221236&md5=bdb7ce74926a6d247eb9a741b02850dc" title="Click to view the MathML source">YN and describe the limiting ⁎-distribution of 2123616303044&_mathId=si139.gif&_user=111111111&_pii=S0022123616303044&_rdoc=1&_issn=00221236&md5=c585460f596fa94ab6e6f9a8ff92f82e" title="Click to view the MathML source">(XN,YN). We develop this approach for related models and give recurrence relations for the limiting ⁎-distribution of heavy Wigner and independent diagonal matrices.